Proper time in higher dimensional gravity.

In GR one of the fundamental postulates is that ##-ds^2 = - g_{\mu \nu} dx^\mu dx^\nu## is interpreted as a the time on the clock of an observer of constant spatial coordinates; a comoving observer. How does this translate to higher dimensional theories of gravity? There one has a higher dimensional metric ##d\sigma^2 = G_{ab} dx^a dx^b## which has ##-ds^2## contained within it. Do we still interpret ##d\sigma^2## as a the clock of a comoving observer?